# Floquet topological phases protected by time glide symmetry

## TL;DR

This paper introduces and classifies Floquet topological phases protected by time glide symmetry, a dynamical symmetry combining reflection and half-period time translation, expanding the understanding of topological phases in driven systems.

## Contribution

The authors present lattice models, derive topological invariants, and classify Floquet topological phases with time glide symmetry using Clifford algebra, highlighting their distinction from static topological crystalline insulators.

## Key findings

- Constructed lattice models of time glide symmetric Floquet topological insulators in 2D and 3D.
- Derived topological invariants from half-period evolution operators and symmetry operators.
- Provided a classification table for Floquet phases with time glide symmetry across dimensions.

## Abstract

We study Floquet topological phases in periodically driven systems that are protected by "time glide symmetry", a combination of reflection and half time period translation. Time glide symmetry is an analog of glide symmetry with partial time translation replacing the partial space translation, and hence, is an intrinsically dynamical symmetry which may be engineered in periodically driven systems by exploiting the controllability of driving. We present lattice models of time glide symmetric Floquet topological insulators in two and three dimensions. The topological numbers characterizing those Floquet topological phases are derived from the half period time evolution operator along with time glide operator. Moreover, we classify Floquet topological phases protected by time glide symmetry in general dimensions using a Clifford algebra approach. The obtained classification table is similar to that for topological crystalline insulators protected by static reflection symmetry, but shows nontrivial entries in different combination of symmetries, which clarifies that time glide symmetric Floquet topological phases are a distinct set of topological phases from topological crystalline insulators. We also classify Floquet topological phases with "time screw symmetry," defined as a two-fold spatial rotation accompanied by half-period time translation.